Let us begin by giving a brief introduction on the tuples of operators appearing in [1]}. A particular class of these tuples were first studied by Grinshpan et al. ([2]}) in connection to multivariate von Neumann inequality. For each \(i, p, q \in \lbrace 1,\ldots ,n\rbrace \)  such that \(p \ne q\) , let
\(\hat{X}_i := \lbrace X \in \mathcal {X}^n(\mathcal {H}): (X_1,\ldots ,X_{i-1},X_{i+1},\ldots ,X_n) \in \mathcal {X}^{n-1}(\mathcal {H})\rbrace .\) 
